What is the integral from 0 to pi/2 of 3sin(y)sqrt(cos(y)) ?

The integral from 0 to pi/2 of 3sin(y)sqrt(cos(y)) is 2

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integral from 0 to pi/2 of 3sin(y)sqrt(cos(y))

\int_{0}^{\frac{π}{2}} 3 sin (y) cos (y) d y

2

Solution steps

\int_{0}^{\frac{π}{2}} 3 sin (y) cos (y) d y

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{0}^{\frac{π}{2}} sin (y) cos (y) d y

Apply u-substitution

: \int_{1}^{0} - u d u

= 3 \cdot \int_{1}^{0} - u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 3 (- \int_{0}^{1} - u d u)

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 (- (- \int_{0}^{1} u d u))

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}

= 3 (- (- [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}))

Simplify

= 3 [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}

Compute the boundaries:

\frac{2}{3}

= 3 \cdot \frac{2}{3}

Simplify

= 2

\int_{- 1}^{1} (2 - x^{2}) - x^{2} d x

\int_{0}^{1} 1 + 2 x d x

\int_{0}^{3} 25 - x^{2} d x

\int_{1}^{5} x + 2 d x

\int_{0}^{x} e^{- y^{2}} d y

What is the integral from 0 to pi/2 of 3sin(y)sqrt(cos(y)) ?
The integral from 0 to pi/2 of 3sin(y)sqrt(cos(y)) is 2